Tag Archives: hyper-inaccessible

Erin Carmody successfully defended her dissertation under my supervision at the CUNY Graduate Center on April 24, 2015, and she earned her Ph.D. degree in May, 2015. Her dissertation follows the theme of killing them softly, proving many theorems of the form: given $\kappa$ with large cardinal property $A$, there is a forcing extension in which $\kappa$ no longer has property $A$, but still has large cardinal property $B$, which is very slightly weaker than $A$. Thus, she aims to enact very precise reductions in large cardinal strength of a given cardinal or class of large cardinals. In addition, as a part of the project, she developed transfinite meta-ordinal extensions of the degrees of hyper-inaccessibility and hyper-Mahloness, giving notions such as $(\Omega^{\omega^2+5}+\Omega^3\cdot\omega_1^2+\Omega+2)$-inaccessible among others.

Erin is also an accomplished artist, who has had art shows of her work in New York, and she has pieces for sale. Much of her work has an abstract or mathematical aspect, while some pieces exhibit a more emotional or personal nature. My wife and I have two of Erin’s paintings in our collection:

Meanwhile, every model of PA does have a definable model of PA plus not Con(PA), since one can use the Henkin model of the left most branch through the tree of attempts to build a complete consistent Henkin theory.

You are objecting to something I am not asserting. Your stronger assertion amounts to the same as what I call the scheme of assertions that L satisfies every individual axiom of ZFC. Of course this is stronger than the arithmetic implication that Con(ZF) implies Con( ZFC). I think we agree about this.

But in ZFC or GBC we are only ensured truth predicates for set models, not proper class models. Meanwhile, in Kelley-Morse set theory, we do have first-order truth predicates for the class models, and thus the argument about L does show Con(ZFC) in KM. See jdh.hamkins.org/km-implies-conzfc.

@SimonHenry There is a little more to it than that, since after all, GBC is finitely axiomatizable, but we don't say that GBC implies Con(GBC) just because in GBC we can prove that the universe itself is a model of GBC. The point is that the argument that exists-a-model implies consistency requires having a satisfaction […]

@AsafKaragila Well, of course the set models are closely related, in that the ZFC model is the $L$ of the ZF model, so whatever extra you are getting is gotten that way, because of the closeness of the models. I don't agree that your way of talking about it is stronger, since you are talking […]

What is shown in the cases you mention is not that the model is a model of ZFC, made as a single statement, but rather the scheme of statements that the model satisfies every individual axiom of ZFC, as a separate statement for each axiom. The difference is between asserting "$L$ is a model of […]

Another way to say it is: forcing equivalence is a local version of isomorphism in the Boolean completions: below every condition in one forcing notion, there is a cone in the Boolean algebra isomorphic to a cone in the other one.